Minimal Two-Body Hamiltonians

• Minimal two-body Hamiltonians are models where interactions are limited to pairs of degrees of freedom, providing a foundation for understanding complex quantum phenomena.
• They are constructed using canonical structures and covariant formulations that enable decomposition into center-of-energy and relative dynamics, often reducing complex systems to effective one-body problems.
• These Hamiltonians support frustration-free ground states and exact solvability, playing a pivotal role in quantum simulation, condensed matter physics, and topological quantum computing.

Minimal two-body Hamiltonians are models in which the interactions involve at most pairs of degrees of freedom (sites, particles, modes), with Hamiltonians composed solely of two-body terms. Serving as foundational constructs across quantum many-body physics, condensed matter, quantum information, and quantum chemistry, minimal two-body Hamiltonians play a central role in the emergence of entanglement, classicality, quantum phases, and computational universality. Their minimality refers to the absence of explicit higher-body couplings—yet, through spectral properties, ground state degeneracy, or effective dynamics, these models often encode complex and rich physical phenomena.

1. Canonical Structures and Covariant Formulation

Minimal two-body Hamiltonians are typically formulated within a constrained algebraic structure, where the canonical variables and Poincaré-invariance dictate the construction and dynamics. In relativistic mechanics, as exemplified by the predictive Hamiltonian formalism (Droz-Vincent, 2010), the canonical phase space comprises four-vectors $q_1, q_2$ (positions) and $p_1, p_2$ (momenta), with the Poisson brackets

$$\{q_a^\alpha, p_{b\beta}\} = \delta_{ab}\delta^\alpha_\beta$$

ensuring manifest covariance. The Hamiltonian for each particle is

$H_a = \frac{1}{2}p_a^2 + V_a$

and, in unipotential models (i.e., $V_1 = V_2 = V$), Poincaré invariance requires that the potential depends only on scalar combinations of kinematic variables, for example

$$p^2, \quad (q_1 - q_2)^2, \quad (p_1 - p_2)^2, \quad [(p_1 - p_2) \cdot P]^2 / P^2$$

where $P = p_1 + p_2$ is the total momentum. Crucially, separate evolution parameters $T_1$ and $T_2$ are associated with each particle, but predictivity requires compatibility conditions for the combined system dynamics.

The key property for separating dynamics is that $H_1$ and $H_2$ commute with $P$ and the total angular momentum $J$, facilitating a decomposition into “center-of-energy” and “relative” variables. This decomposition is the backbone for subsequent analyses of energy conservation, orbit structure, and reduction to effective one-body problems.

2. Center-of-Energy and Conservation Laws

In relativistic minimal two-body Hamiltonians, the proper notion of the center-of-mass is replaced by the center-of-energy, reflecting the Lorentz covariance and associated weights: $E = \frac{M_1 x_1 + M_2 x_2}{M_1 + M_2}$ with $M_a = P \cdot p_a/|P|$ and $E_a$ (individual “energies”) separately conserved thanks to translation invariance and the unipotential structure. This structure ensures that, in the classical regime where both $M_a$ are positive, the location $E$ is analogous to the Newtonian center-of-mass, but constructed from conserved energies rather than rest masses.

For quantum systems such as two anyons in planar geometry (Correggi et al., 2018), the minimal Hamiltonian takes the form

$$H_\alpha = -\left(i\nabla_1 + \alpha A(x_1 - x_2)\right)^2 - \left(i\nabla_2 + \alpha A(x_2 - x_1)\right)^2 + V(|x_1 - x_2|)$$

and self-adjoint extension theory (via quadratic forms) is used to classify all physically admissible boundary conditions, leading to closedness, boundedness from below, and the emergence of a one-parameter family of minimal extensions characterized by boundary behavior in the s-wave channel.

3. Relative Motion and Effective One-Body Reduction

A crucial feature—especially in the relativistic context—is that the equations of motion for “internal” (relative) variables $(z = q_1 - q_2,\, y = p_1 - p_2)$ are isomorphic to those for a nonrelativistic one-body system, but with a nonaffine time parameter. The effective internal Hamiltonian is

$H_{int} = \frac{1}{2}y^2 + V_{eff}(z, y)$

and the dynamical equations in the total evolution parameter $\lambda = T_1 + T_2$ are

$$\frac{dz}{d\lambda} = \{z, H_{int}\}, \quad \frac{dy}{d\lambda} = \{y, H_{int}\}$$

However, unless the motion is circular and certain Poisson bracket conditions are satisfied (e.g., $\{z^2, V\} = 0$ and $\{z \cdot y, V\} = 0$), the schedule relating $\lambda$ to the center-of-mass time $T$ is non-linear. Only in the circular case does the analogy with nonrelativistic periodic motion become exact.

In the nonrelativistic or algebraically structured models—typified by Jastrow parent Hamiltonians (Beau et al., 2021)—the construction follows a product ansatz for the wave function,

$\Psi_0 = \prod_{i < j} f(r_{ij})$

and the application of the kinetic term generates minimal two-body ($V_2$) and, in general, three-body ($V_3$) potentials: $$V_2 = \frac{\hbar^2}{m}\sum_{i<j}\left[\frac{f''(r_{ij})}{f(r_{ij})} + (d-1)\frac{f'(r_{ij})}{r_{ij}f(r_{ij})}\right]$$ with $V_3$ directly expressible in terms of products and derivatives of $f(r)$ and geometrical factors ($\vec{r}_{ij}/r_{ij}$, etc.).

4. Ground State Structure, Frustration-Freeness, and Exact Solvability

Minimal two-body Hamiltonians often support a ground state subspace with highly structured entanglement and degeneracy. For two-body frustration-free models—for instance, parent Hamiltonians for Matrix Product States (MPS) or qubit spin models (Ji et al., 2010, Schuch et al., 13 Mar 2025)—the entire ground space forms the span of a fixed tree tensor network (TTN). Formally, if

$H = \sum_J H_J$

(summed over two-body terms), then the ground space is

$$\mathcal{K}(H) = \bigcap_J (\mathcal{K}(H_J) \otimes \mathcal{H}_{\overline{J}})$$

In these systems, degeneracy counting reduces to a combinatorial problem (in the class $\#P$), and entangled ground states only arise in the presence of ground space degeneracy.

For continuous-variable quantum computation (Aolita et al., 2010), quadratic two-body Hamiltonians of the form

$$H_G(s) = \sum_{i \in \mathcal{V}}\frac{\omega_i}{2}\left(\hat{q}_i^2/s^4 + \hat{N}_i^2\right)$$

generate gapped, frustration-free systems with Gaussian graph states as their unique ground states. The nullifier operators $\hat{N}_i$ enforce the stabilizer structure, and the presence of a constant energy gap (proportional to $1/s^2$) provides robust resource states for measurement-based quantum computation.

In minimal Hamiltonians designed for topological order, such as toric codes and quantum double models (Brell et al., 2010), systematic use of perturbation gadgets allows for the exact realization of multi-body parent Hamiltonians as low-energy effective models of strictly two-body systems. By engineering error-detecting subsystem codes within small code gadgets, one ensures that perturbative processes that contribute to the effective low-energy Hamiltonian precisely mimic the original higher-body constraints, thus faithfully reproducing both the operator algebra and symmetries of the underlying topological phase.

5. Minimality, Approximations, and Physical Realizability

While some phenomena (for example, quantum Hall phases such as the PH-Pfaffian (Pakrouski, 2021)) require higher-body interaction terms for exact ground state realization, practical constraints and theoretical minimality principles motivate the search for two-body approximate parent Hamiltonians. Optimization techniques—in the space of pseudopotentials for quantum Hall systems, or via deformation operators in the context of MBQC resource states—allow for the engineering of minimal two-body Hamiltonians with ground states of arbitrarily high fidelity to the target state, provided certain trade-offs are accepted (e.g., shrinking spectral gap or exponential ground space degeneracy in the limiting case).

In quantum simulation, the universality of digital-analog architectures (Garcia-de-Andoin et al., 2023) can be leveraged in arbitrary two-body Hamiltonian settings, where resources are organized as alternations of analog blocks (evolved under the two-body source Hamiltonian) and fast single-qubit operations, with efficient protocols guaranteeing simulation of arbitrary two-body target Hamiltonians in $O(n^2)$ analog blocks (for $n$ qubits), outperforming earlier $O(n^3)$ constructions.

In the construction of contact (zero-range) interactions for $n > 2$ in three spatial dimensions (Ferretti et al., 9 Jul 2024), the notion of minimality is intrinsically tied to boundary conditions on the coincidence hyperplanes, with the physical requirement of stability necessitating the inclusion of position-dependent (multi-particle) modifications to purely two-body contact conditions. This regularization removes ultraviolet divergences and prevents “fall to the center” instabilities characteristic of naive two-body zero-range models for $n \geq 3$.

6. Reduction to One-Body Problems and Limiting Cases

Minimal two-body Hamiltonians facilitate the reduction of complex systems to effective one-body problems under suitable limits. In the extreme mass ratio (e.g., $m_1 \ll m_2$) limit of covariant two-body dynamics (Droz-Vincent, 2010), the heavy body's worldline and the center-of-energy coincide (provided the binding energy remains “not too large”), justifying the effective external field approximation for the dynamics of the light particle. This result mirrors the reduction in impurity models in zero-range interaction systems (Ferretti et al., 9 Jul 2024): the limit $M \to \infty$ converts a many-body Hamiltonian of $N$ distinguishable heavy particles and an impurity into a one-body Hamiltonian for the light particle subject to $N$ fixed but nonlocal point interactions, free from the ultraviolet pathologies that plague standard contact interaction models.

In harmonic oscillator models (Willemyns et al., 2021), analytic diagonalization under specific mass-coupling conditions diagonalizes the two-body interaction block, yielding a set of decoupled normal modes and analytical energy formulas. This facilitates explicit solution of translation-invariant $N$-body systems with purely two-body interactions for arbitrary mass arrangements, provided the coupling constants satisfy prescribed relations.

7. Trends in Complexity, Simulation, and Entanglement Structure

Minimal two-body Hamiltonians are often the benchmark for understanding the complexity of many-body quantum states and their simulability. Ground and thermal states of strictly two-body Hamiltonians (the “quantum exponential family” (Huber et al., 2016)) are exactly those states fully specified by their two-body marginals. States outside the convex hull of this set (i.e., conv($\mathcal{Q}_2$)) necessarily have irreducible $k$-body correlations ($k > 2$) and cannot be engineered without explicit higher-body terms.

Certification methods—including witness operators—enable experimental discrimination between states preparable by minimal two-body Hamiltonians and those which fundamentally require high-order correlations. Complexity-theoretic correspondence is seen in the mapping between ground space degeneracies of two-body frustration-free Hamiltonians and classical counting problems (e.g., #2-SAT) (Ji et al., 2010), further justifying the classical flavor of entanglement in these minimal models.

In quantum information and condensed matter, the paper and extension of minimal two-body Hamiltonians underpin advances in scalable quantum architectures, ground state engineering, and the theoretical understanding of the boundary between classical and quantum complexity across many platforms.

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This article synthesizes the core technical structures, mathematical principles, and physical implications emerging from the arXiv literature on minimal two-body Hamiltonians, with representative citations from (Droz-Vincent, 2010, Ji et al., 2010, Brell et al., 2010, Beau et al., 2021, Pakrouski, 2021, Garcia-de-Andoin et al., 2023, Ferretti et al., 9 Jul 2024, Schuch et al., 13 Mar 2025) and related works.